Wiener's theorem for positive definite functions on hypergroups

The following theorem on the circle group $\mathbb{T}$ is due to Norbert Wiener: If $f\in L^{1}\left( \mathbb{T}\right)$ has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then $f\in L^{2}\left( \mathbb{T}\right)$. This result has been extended to even exponents including $p=\infty$, but shown to fail for all other $p\in\left( 1,\infty\right] .$ All of this was extended further (appropriately formulated) well beyond locally compact abelian groups. In this paper we prove Wiener's theorem for even exponents for a large class of commutative hypergroups. In addition, we present examples of commutative hypergroups for which, in sharp contrast to the group case, Wiener's theorem holds for all exponents $p\in\left[ 1,\infty\right]$. For these hypergroups and the Bessel-Kingman hypergroup with parameter $\frac{1}{2}$ we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces.